Semi-classical states of nonlinear Schrödinger equations: a variational reduction method
type: Item
instance of: scholarly article
title: Semi-classical states of nonlinear Schrödinger equations: a variational reduction method
zbMATH Open document ID: 1030.35031
author: P. Felmer
author: Manuel del Pino
published in: Mathematische Annalen
publication date: 2002-12-01T00:00:00Z
full work available at URL: http://hdl.handle.net/10533/173686
review text: For a smooth, bounded and uniformly positive potential \(V\), the authors prove the existence of solutions \(v_\varepsilon\) of the nonlinear Schrödinger equation \[ \Delta v_\varepsilon-V(P+\varepsilon y)v_\varepsilon+f(v_\varepsilon)=0, \qquad y\in{\mathbb R}^N, \] vanishing as \(|x|\to\infty\) which approaches a positive ground-state solution \(w\) of the nonlinear Schrödinger equation \[ \Delta w-V(P)w+f(w)=0,\qquad y\in{\mathbb R}^N, \] vanishing at infinity and radially symmetry about some point, under the following hypotheses: (\(f_0\)) \(f(s)\) is of class \(C^1\) and \(f^\prime(s)\) is Lipschitz on \([0,\infty)\), (\(f_1\)) \(\exists p\in(1,{{N+2}\over{N-2}})\) if \(N\geq 3\) such that \(f^\prime(s)=O(s^{p-1})\) as \(s\to+\infty\), (\(f_2\)) \(sf^\prime(s)/f(s)\) is bounded on \((0,1)\), and (\(f_3\)) \(\exists q>1\) such that \(0<qf(s)\leq sf^\prime(s)\) for all \(s>0\). Under these general hypotheses it is left unanswered if the above \(w\) is unique up to translation. A variational method of constructing single and multi-spike solutions, associated with general topologically nontrivial critical points of \(V\), is developed.
Mathematics Subject Classification ID: 35J20
Mathematics Subject Classification ID: 35J10
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35B25
Mathematics Subject Classification ID: 58E05
zbMATH DE Number: 1837347
zbMATH Keywords: nonlinear Schödinger equation
zbMATH Keywords: variational method
zbMATH Keywords: multi-spike solution
MaRDI profile type: Publication
OpenAlex ID: W1977632986
DOI: 10.1007/S002080200327
description: scientific article; zbMATH DE number 1837347